Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process $\langle X,Y \rangle_t$ for all $t \geq 0$.
My idea is (since $X_t $ and $Y_t$ are cont. local martingales) to use that
$$\langle X,Y \rangle_t = \langle (W_s^2 \cdot W_s ), (1 \cdot W_t^7) \rangle_t = \int_0^t W_s^2 \mathrm{d} \langle W,W^7 \rangle_s $$
But I don´t know how to calculate the integral above. Should I find $\langle W,W^7 \rangle_t,$ where I use that $\langle W,W^7 \rangle_t,$ makes $W_tW_t^7 - \langle W,W^7 \rangle_t, = W_t^8 - \langle W,W^7 \rangle_t$ a continuous local martingale?